Optimal. Leaf size=47 \[ \frac{\text{li}\left (c \left (b x^2+a\right )\right )}{2 b c}-\frac{a+b x^2}{2 b \log \left (c \left (a+b x^2\right )\right )} \]
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Rubi [A] time = 0.0430946, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2454, 2389, 2297, 2298} \[ \frac{\text{li}\left (c \left (b x^2+a\right )\right )}{2 b c}-\frac{a+b x^2}{2 b \log \left (c \left (a+b x^2\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2389
Rule 2297
Rule 2298
Rubi steps
\begin{align*} \int \frac{x}{\log ^2\left (c \left (a+b x^2\right )\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\log ^2(c (a+b x))} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\log ^2(c x)} \, dx,x,a+b x^2\right )}{2 b}\\ &=-\frac{a+b x^2}{2 b \log \left (c \left (a+b x^2\right )\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{\log (c x)} \, dx,x,a+b x^2\right )}{2 b}\\ &=-\frac{a+b x^2}{2 b \log \left (c \left (a+b x^2\right )\right )}+\frac{\text{li}\left (c \left (a+b x^2\right )\right )}{2 b c}\\ \end{align*}
Mathematica [A] time = 0.0192256, size = 43, normalized size = 0.91 \[ \frac{\frac{\text{li}\left (c \left (b x^2+a\right )\right )}{c}-\frac{a+b x^2}{\log \left (c \left (a+b x^2\right )\right )}}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.07, size = 59, normalized size = 1.3 \begin{align*} -{\frac{{x}^{2}}{2\,\ln \left ( c \left ( b{x}^{2}+a \right ) \right ) }}-{\frac{a}{2\,b\ln \left ( c \left ( b{x}^{2}+a \right ) \right ) }}-{\frac{{\it Ei} \left ( 1,-\ln \left ( c \left ( b{x}^{2}+a \right ) \right ) \right ) }{2\,bc}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b x^{2} + a}{2 \,{\left (b \log \left (b x^{2} + a\right ) + b \log \left (c\right )\right )}} + \int \frac{x}{\log \left (b x^{2} + a\right ) + \log \left (c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94823, size = 130, normalized size = 2.77 \begin{align*} -\frac{b c x^{2} + a c - \log \left (b c x^{2} + a c\right ) \logintegral \left (b c x^{2} + a c\right )}{2 \, b c \log \left (b c x^{2} + a c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.80688, size = 49, normalized size = 1.04 \begin{align*} \begin{cases} \frac{x^{2}}{2 \log{\left (a c \right )}} & \text{for}\: b = 0 \\0 & \text{for}\: c = 0 \\\frac{\operatorname{Ei}{\left (\log{\left (a c + b c x^{2} \right )} \right )}}{2 b c} & \text{otherwise} \end{cases} + \frac{- a - b x^{2}}{2 b \log{\left (c \left (a + b x^{2}\right ) \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27656, size = 61, normalized size = 1.3 \begin{align*} -\frac{\frac{b c x^{2} + a c}{\log \left ({\left (b x^{2} + a\right )} c\right )} -{\rm Ei}\left (\log \left ({\left (b x^{2} + a\right )} c\right )\right )}{2 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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